arXiv subjects

P. Krupski

Publications and source records attributed to P. Krupski.

3 recordsLinked to original sources

arXiv2023

On generalized $V^n$-continua

The notion of a $V^n$-continuum was introduced by Alexandroff \cite{ps} as a generalization of the concept of $n$-manifold. In this note we consider the cohomological analogue of $V^n$-continuum and prove that any strongly locally homogeneous generalized continuum $X$ with cohomological dimension $\dim_G X=n$ is a generalized $V^n$-space with respect to the cohomological dimension $\dim_G$. In particular, every strongly locally homogeneous continuum of covering dimension $n$ is a $V^n$-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in \cite{tv}. An analog of the Mazurkiewicz theorem that no subset of covering dimension $\le n-2$ cuts any region of the Euclidean $n$-space is also obtained for strongly locally homogeneous generalized continua $X$ of cohomological dimension $\dim_G X=n$.

math.GN↗

arXiv2009

Mazurkiewicz manifolds and homogeneity

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an $F_σ$-subset of a "smaller" dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces.

math.GN↗

arXiv2008

Generalized Cantor manifolds and homogeneity

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class $\mathcal C$ is a strong Cantor manifold (or at least a Cantor manifold) with respect to $\mathcal C$. Here, the class $\mathcal C$ is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered.

math.GN↗